International Journal of Mathematics and Mathematical Sciences
Volume 2 (1979), Issue 4, Pages 693-701
doi:10.1155/S0161171279000521

A distributional Hardy transformation

R. S. Pathak1 and J. N. Pandey2

1Department of Mathematics, Banaras Hindu University, Varanasi, India
2Department of Mathematics, Carleton University, Ottawa, Canada

Received 23 January 1979

Copyright © 1979 R. S. Pathak and J. N. Pandey. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The Hardy's F-transform F(t)=∫0∞Fv(ty)yf(y)dy is extended to distributions. The corresponding inversion formula f(x)=∫0∞Cv(tx)tF(t)dt is shown to be valid in the weak distributional sense. This is accomplished by transferring the inversion formula onto the testing function space for the generalized functions under consideration and then showing that the limiting process in the resulting formula converges with respect to the topology of the testing function space.